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Concept
Hurewicz theorem
Summary
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results of Henri Poincaré.
The Hurewicz theorems are a key link between homotopy groups and homology groups.
For any path-connected space X and positive integer n there exists a group homomorphism
called the Hurewicz homomorphism, from the n-th homotopy group to the n-th homology group (with integer coefficients). It is given in the following way: choose a canonical generator , then a homotopy class of maps is taken to .
The Hurewicz theorem states cases in which the Hurewicz homomorphism is an isomorphism.
For , if X is -connected (that is: for all ), then for all , and the Hurewicz map is an isomorphism. This implies, in particular, that the homological connectivity equals the homotopical connectivity when the latter is at least 1. In addition, the Hurewicz map is an epimorphism in this case.
For , the Hurewicz homomorphism induces an isomorphism , between the abelianization of the first homotopy group (the fundamental group) and the first homology group.
For any pair of spaces and integer there exists a homomorphism
from relative homotopy groups to relative homology groups. The Relative Hurewicz Theorem states that if both and are connected and the pair is -connected then for and is obtained from by factoring out the action of . This is proved in, for example, by induction, proving in turn the absolute version and the Homotopy Addition Lemma.
This relative Hurewicz theorem is reformulated by as a statement about the morphism
where denotes the cone of . This statement is a special case of a homotopical excision theorem, involving induced modules for (crossed modules if ), which itself is deduced from a higher homotopy van Kampen theorem for relative homotopy groups, whose proof requires development of techniques of a cubical higher homotopy groupoid of a filtered space.
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Ontological neighbourhood
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Related concepts (11)
Homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry (e.g., A1 homotopy theory) and (specifically the study of ). In homotopy theory and algebraic topology, the word "space" denotes a topological space.
Simplicial set
In mathematics, a simplicial set is an object composed of simplices in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and . Formally, a simplicial set may be defined as a contravariant functor from the to the . Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber. Every simplicial set gives rise to a "nice" topological space, known as its geometric realization.
Homotopical connectivity
In algebraic topology, homotopical connectivity is a property describing a topological space based on the dimension of its holes. In general, low homotopical connectivity indicates that the space has at least one low-dimensional hole. The concept of n-connectedness generalizes the concepts of path-connectedness and simple connectedness. An equivalent definition of homotopical connectivity is based on the homotopy groups of the space. A space is n-connected (or n-simple connected) if its first n homotopy groups are trivial.